NegativelyOrientedPoints
✖
NegativelyOrientedPoints
tests whether the sequence of points p1,p2,p3,…,pn is negatively oriented.
Details

- NegativelyOrientedPoints is also known as clockwise in 2D and left‐hand rule in 3D.
- Typically used to determine the orientation of a rotational motion with respect to a set of points.
- In two dimensions, NegativelyOrientedPoints[{p1,p2,p3}] gives True if the point p3 is in the half-plane bounded by the line through p1 and p2 and extended in the direction of {1,0}.
- For negatively oriented points p1, p2 and p3, the determinant of the matrix {p2-p1, p3-p1} is negative.
- In three dimensions, NegativelyOrientedPoints[{p1,p2,p3,p4}] gives True if the point p4 is in the half-space bounded by the plane through the point p1 with normal direction (p3-p1)(p2-p1).
- For negatively oriented points p1, p2, p3 and p4, the dot product of p4-p1 and (p3-p1)(p2-p1) is negative.
- In d dimensions, d+1 points p1,p2,…,pd+1 are negatively oriented if the determinant of the matrix {p2-p1,…,pd+1-p1} is negative.


Examples
open allclose allBasic Examples (2)Summary of the most common use cases
The points {0,0}, {1,1}, {.5,-1} are negatively oriented:

https://wolfram.com/xid/0b7f618tcp4aq4wm-7f2o6y

https://wolfram.com/xid/0b7f618tcp4aq4wm-7kqtw5


https://wolfram.com/xid/0b7f618tcp4aq4wm-ofnazw

Find the condition for which a point is below a plane:

https://wolfram.com/xid/0b7f618tcp4aq4wm-5nyoxu

Scope (3)Survey of the scope of standard use cases
NegativelyOrientedPoints works with two-dimensional points:

https://wolfram.com/xid/0b7f618tcp4aq4wm-o520tf


https://wolfram.com/xid/0b7f618tcp4aq4wm-d8st3j

NegativelyOrientedPoints works with numerical coordinates:

https://wolfram.com/xid/0b7f618tcp4aq4wm-vaa7iv


https://wolfram.com/xid/0b7f618tcp4aq4wm-loksgh

NegativelyOrientedPoints over a set of coordinates:

https://wolfram.com/xid/0b7f618tcp4aq4wm-iosq0l


https://wolfram.com/xid/0b7f618tcp4aq4wm-pbm87t


https://wolfram.com/xid/0b7f618tcp4aq4wm-odxneo

Generalizations & Extensions (1)Generalized and extended use cases
Give assumptions to NegativelyOrientedPoints:

https://wolfram.com/xid/0b7f618tcp4aq4wm-2lmdu

https://wolfram.com/xid/0b7f618tcp4aq4wm-hnpi36


https://wolfram.com/xid/0b7f618tcp4aq4wm-zuzvxg

Applications (5)Sample problems that can be solved with this function
Basic Applications (2)
Graph negatively oriented points:

https://wolfram.com/xid/0b7f618tcp4aq4wm-0e61c

https://wolfram.com/xid/0b7f618tcp4aq4wm-x5hix


https://wolfram.com/xid/0b7f618tcp4aq4wm-p801bp


https://wolfram.com/xid/0b7f618tcp4aq4wm-bb3lo1

https://wolfram.com/xid/0b7f618tcp4aq4wm-c5qez8


https://wolfram.com/xid/0b7f618tcp4aq4wm-qklxhp

Geometry (3)
Faces of a polyhedron are positively oriented:

https://wolfram.com/xid/0b7f618tcp4aq4wm-ota0wc

https://wolfram.com/xid/0b7f618tcp4aq4wm-pvuukh


https://wolfram.com/xid/0b7f618tcp4aq4wm-c9yy0i

NegativelyOrientedPoints over lines in 2D:

https://wolfram.com/xid/0b7f618tcp4aq4wm-9jvrr0

It is equivalent to the orientation of the consecutive vertices of the line:

https://wolfram.com/xid/0b7f618tcp4aq4wm-566q3k

Show the robustness of NegativelyOrientedPoints:

https://wolfram.com/xid/0b7f618tcp4aq4wm-s8g3om

https://wolfram.com/xid/0b7f618tcp4aq4wm-o3ft7h

Properties & Relations (4)Properties of the function, and connections to other functions
NegativelyOrientedPoints returns False for collinear points:

https://wolfram.com/xid/0b7f618tcp4aq4wm-qoz0qg

https://wolfram.com/xid/0b7f618tcp4aq4wm-ek0gp9


https://wolfram.com/xid/0b7f618tcp4aq4wm-d0hir4

NegativelyOrientedPoints returns False if positively oriented:

https://wolfram.com/xid/0b7f618tcp4aq4wm-8gvtvm

https://wolfram.com/xid/0b7f618tcp4aq4wm-isc17j


https://wolfram.com/xid/0b7f618tcp4aq4wm-wxx8sf

Use RegionMember to test whether points are negatively oriented:

https://wolfram.com/xid/0b7f618tcp4aq4wm-j0931q


https://wolfram.com/xid/0b7f618tcp4aq4wm-m9ot9p

3D points are coplanar if they are neither positively nor negatively oriented:

https://wolfram.com/xid/0b7f618tcp4aq4wm-xr9x2n

https://wolfram.com/xid/0b7f618tcp4aq4wm-cq6slx


https://wolfram.com/xid/0b7f618tcp4aq4wm-2gfk11

Wolfram Research (2020), NegativelyOrientedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.
Text
Wolfram Research (2020), NegativelyOrientedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.
Wolfram Research (2020), NegativelyOrientedPoints, Wolfram Language function, https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.
CMS
Wolfram Language. 2020. "NegativelyOrientedPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.
Wolfram Language. 2020. "NegativelyOrientedPoints." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html.
APA
Wolfram Language. (2020). NegativelyOrientedPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html
Wolfram Language. (2020). NegativelyOrientedPoints. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html
BibTeX
@misc{reference.wolfram_2025_negativelyorientedpoints, author="Wolfram Research", title="{NegativelyOrientedPoints}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html}", note=[Accessed: 10-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_negativelyorientedpoints, organization={Wolfram Research}, title={NegativelyOrientedPoints}, year={2020}, url={https://reference.wolfram.com/language/ref/NegativelyOrientedPoints.html}, note=[Accessed: 10-July-2025
]}